Proof of generalization of a particular limit converging to $e^{\frac{1}{(p-1)^2}}$

Question

I was reading a very old and long article on logarithms in a library it has pages turned yellow and had one pages titled - Tricky problems I managed to solve 5 out of the 6 but I couldn't do this 6th one . Prove that for all $ p \in \mathbb{N}$ - $$\lim_{n \rightarrow \infty} \frac{(1^{1^p} \times 2^{2^p} \times 3^{3^p} \times \ldots n^{n ^p} )}{n^{\frac{1}{p+1}}} =e^{\frac{1}{(p-1)^2}}$$ I tried to manupilate this with the properties of logarithms but failed.

Answer 1

It seems to be wrong. The logarithm of the lhs is $$\text{HurwitzZeta}^{(1,0)}(-p,n+1)-\frac{\log (n)}{p+1}-\zeta '(-p)$$ which goes to infinity with $n$